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Enumerative lattice algorithms in any norm via m-ellipsoid coverings
- in FOCS
, 2011
"... We give a novel algorithm for enumerating lattice points in any convex body, and give applications to several classic lattice problems, including the Shortest and Closest Vector Problems (SVP and CVP, respectively) and Integer Programming (IP). Our enumeration technique relies on a classical concept ..."
Abstract
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Cited by 18 (13 self)
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to the lattice, on n-dimensional lattices in any (semi-)norm given an M-ellipsoid of the unit ball. In many norms of interest, including all ℓp norms, an M-ellipsoid is computable in deterministic poly(n) time, in which case these algorithms are fully deterministic. Here our approach may be seen as a
Enumerative Algorithms for the Shortest and Closest Lattice Vector Problems in Any Norm via M-Ellipsoid Coverings
, 2010
"... We give an algorithm for solving the exact Shortest Vector Problem in n-dimensional lattices, in any norm, in deterministic 2 O(n) time (and space), given poly(n)-sized advice that depends only on the norm. In many norms of interest, including all ℓp norms, the advice is efficiently and deterministi ..."
Abstract
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Cited by 5 (1 self)
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is to reduce the enumeration of lattice points in an arbitrary convex body K to enumeration in 2 O(n) copies of an M-ellipsoid of K, a classical concept in asymptotic convex geometry. Building on the techniques of Klartag (Geometric and Functional Analysis, 2006), we also give an expected 2 O(n)-time algorithm
Deterministic Construction of an Approximate M-Ellipsoid and its Applications to Derandomizing Lattice Algorithms
, 2012
"... We give a deterministic O(log n) n-time and space algorithm for the Shortest Vector Problem (SVP) of a lattice under any norm, improving on the previous best deterministic nO(n)-time algorithms for general norms. This approaches the 2O(n)-time and space complexity of the randomized sieve based SVP a ..."
Abstract
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Cited by 3 (2 self)
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algorithms (Arvind and Joglekar, FSTTCS 2008), first introduced by Ajtai, Kumar and Sivakumar (STOC 2001) for ℓ2-SVP, and the M-ellipsoid covering based SVP algorithm of Dadush et al. (FOCS 2011). Here we continue with the covering approach of Dadush et al., and our main technical contribution is a
A Deterministic Polynomial Space Construction for ɛ-nets under any Norm
, 2013
"... We give a deterministic polynomial space construction for nearly optimal -nets with respect to any input n-dimensional convex body K and norm ‖ · ‖. More precisely, our algorithm can build and iterate over an -net of K with respect to ‖ · ‖ in time 2O(n) × ( size of the optimal net) using only ..."
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in these lattices can be efficiently performed using poly(n)-space. This also yields the first existential construction of poly(n)-space enumerable thin covering lattices for general convex bodies, which we believe is of inde-pendent interest. Our construction combines the use of the M-ellipsoid from convex
APractical Lattice-based Digital Signature Schemes
"... Digital signatures are an important primitive for building secure systems and are used in most real world security protocols. However, almost all popular signature schemes are either based on the factoring as-sumption (RSA) or the hardness of the discrete logarithm problem (DSA/ECDSA). In the case o ..."
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of classical cryptanalytic advances or progress on the development of quantum computers the hardness of these closely related problems might be seriously weakened. A potential alternative approach is the construction of sig-nature schemes based on the hardness of certain lattices problems which are assumed
Akademisk avhandling för teknisk doktorsexamen vid
, 1994
"... mcmxciv This thesis deals with combinatorics in connection with Coxeter groups, finitely generated but not necessarily finite. The representation theory of groups as nonsingular matrices over a field is of immense theoretical importance, but also basic for computational group theory, where the group ..."
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the group elements are data structures in a computer. Matrices are unnecessarily large structures, and part of this thesis is concerned with small and efficient representations of a large class of Coxeter groups (including most Coxeter groups that anyone ever payed any attention to.) The main contents
Physics
, 2011
"... Ultracold atoms in optical lattices provide a highly control-lable environment for the clean experimental realization of various model Hamiltonians from condensed matter and statistical physics. For example, the two-component Bose-Hubbard model, which re-duces to an anisotropic spin-1/2 Heisenberg m ..."
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Ultracold atoms in optical lattices provide a highly control-lable environment for the clean experimental realization of various model Hamiltonians from condensed matter and statistical physics. For example, the two-component Bose-Hubbard model, which re-duces to an anisotropic spin-1/2 Heisenberg
POUR L'OBTENTION DU GRADE DE DOCTEUR ÈS SCIENCES PAR
"... 2010 to my wife, Joyce, and my family...- Résumé- ..."
Results 1 - 10
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